Physical Phenomenon and Mathematical Basic Knowledge
A background theory which is necessary to understand both of a conventional technique and the present invention will be described. First, a physical phenomenon necessary for the description of the conventional technique will be described. When an object moves in a magnetic field changing as the time passes, electromagnetic induction generates two types of electric fields, namely (a) electric field E(i)=∂A/∂t, which is generated by a time-change of a magnetic field, and (b) electric field E(v)=v×B, which is generated as the object moves in the magnetic field. In this case, v×B represents the outer product of v and B, ∂A/∂t represents the partial differential of A with respect to time. In this case, v, B, and A respectively correspond to the following and are vectors having directions in three dimensions (x, y, and z) (v: flow velocity, B: magnetic flux density, and A: vector potential (whose relationship with the magnetic flux density is represented by B=rotA)). Note, however, that the three-dimensional vectors in this case differ in meaning from vectors on a complex plane. These two types of electric fields generate a potential distribution in the fluid, and electrodes can detect this potential.
Generally known mathematical basic knowledge will be described next. A cosine wave P·cos(ω·t) and a sine wave Q·sin(ω·t) which have the same frequency but different amplitudes are combined into the following cosine wave. In this case, P and Q are amplitudes, and ω is an angular frequency.P·cos(ω·t)+Q·sin(ω·t)=(P2+Q2)1/2·cos(ω·t−ε) where ε=tan−1(Q/P)  (1)
In order to analyze the combining operation in equation (1), it is convenient to perform mapping on a complex coordinate plane so as to plot an amplitude P of cosine wave P·cos(ω·t) along a real axis and an amplitude Q of the sine wave Q·sin(ω·t) along an imaginary axis. That is, on the complex coordinate plane, a distance (P2+Q2)1/2 from the origin gives the amplitude of the combined wave, and an angle ε=tan−1(Q/P) gives the phase difference between the combined wave and ω·t.
In addition, on the complex coordinate plane, the following relational expression holds.L·exp(j·ε)=L·cos(ε)+j·L·sin(ε)  (2)Equation (2) is an expression associated with a complex vector, in which j is an imaginary unit, L gives the length of the complex vector, ands gives the direction of the complex vector. In order to analyze the geometrical relationship on the complex coordinate plane, it is convenient to use conversion to a complex vector.
The following description uses mapping onto a complex coordinate plane like that described above and geometrical analysis using complex vectors to show how an inter-electrode electromotive force behaves and describe how the conventional technique uses this behavior.
Description about Conventional Technique
A complex vector arrangement of inter-electrode electromotive force generated with one coil set and an electrode pair in the electromagnetic flowmeter (see Japanese Patent No. 3774218 (Patent Document 1), Japanese Patent Application Publication No. 2005-300325 (Patent Document 2), and Japanese Patent Application Publication No. 2005-300326 (Patent Document 3) ) suggested by the inventor will be described next. FIG. 32 is a block diagram illustrating a configuration of the electromagnetic flowmeter disclosed in Patent Document 1 to Patent Document 3. This electromagnetic flowmeter includes a measuring tube 1 through which a fluid to be measured flows, a pair of electrodes 2a, 2b which are placed to face each other in the measuring tube 1 so as to be perpendicular to both a magnetic field to be applied to the fluid to be measured and an axis PAX of the measuring tube 1 and come into contact with the fluid to be measured, and detect the electromotive force generated by the magnetic flow and the flow of the fluid to be measured, an exciting coil 3 which applies, to the fluid to be measured, a time-changing magnetic field asymmetric on the front and rear sides of the measuring tube 1 which are bordered on a plane PLN, perpendicular to the direction of the axis PAX of the measuring tube, which includes the electrodes 2a, 2b, with the plane PLN serving as a boundary of the measuring tube 1, a power supply unit 4 supplying an exciting current to the exciting coil 3 to generate a magnetic field, a signal conversion unit 5 detecting an electromotive force between the electrodes 2a, 2b, and a flow rate output unit 6 calculating the flow rate of the fluid to be measured based on the inter-electrode electromotive force detected by the signal conversion unit 5, and signal lines 7a, 7b connecting between the signal conversion unit 5 and the electrodes 2a, 2b. 
When the exciting coil 3 applies, to the fluid to be measured, a magnetic field asymmetric on the front and rear sides of the measuring tube 1 which are bordered on the plane PLN including the electrodes 2a, 2b with the plane PLN serving as the boundary of the measuring tube 1, the vector mapped on a complex plane based on the amplitudes of an measured inter-electrode electromotive force and a phase difference corresponds to the resultant vector Va+Vb of the vector Va of the ∂A/∂t component and the vector Vb of the v×B component.Va=B·rω·exp(j·θω)·ω  (3)Vb=B·rv·exp(j·θv)·V  (4)
FIG. 33 shows the vectors Va and Vb. In FIG. 33, Re represents a real axis, and Im represents an imaginary axis. The vector Va of the ∂A/∂t component is the electromotive force generated by a change in magnetic field, and hence has a magnitude proportional to an exciting angular frequency ω. In this case, let B be the magnitude of a magnetic field, rω be a known proportional constant portion other than a magnitude of a magnetic field corresponding to the magnitude of the vector Va, and θω be the direction of the vector Va. In addition, the vector Vb of the v×B component is the electromotive force generated by the movement of the fluid to be measured in the measuring tube, and hence has a magnitude proportional to the magnitude V of the flow velocity. In this case, let rv be a known proportional constant portion other than a magnitude of a magnetic field corresponding the magnitude of the vector Vb, and θv be the direction of the vector.
Based on the complex vector arrangement as shown in FIG. 33 and equations (3) and (4), the electromagnetic flowmeter disclosed in Patent Document 1 to Patent Document 3 extracts a parameter (asymmetric excitation parameter) free from the influence of a span shift, and outputs a flow rate based on the asymmetric excitation parameter, thus solving the problem of the span shift.
A span shift will be described with reference to FIG. 34. Assume that the magnitude V of the flow velocity measured by the electromagnetic flowmeter has changed in spite of the fact that the flow velocity of a fluid to be measured has not changed. In such a case, a span shift can be considered as a cause of this output variation. For example, assume that calibration is performed such that when the flow velocity of a fluid to be measured is 0 in an initial state, the output from the electromagnetic flowmeter becomes 0 (v), and when the flow velocity is 1 (m/sec), the output becomes 1 (v). In this case, an output from the electromagnetic flowmeter is a voltage representing the magnitude V of a flow velocity. According to this calibration, if the flow velocity of a fluid to be measured is 1 (m/sec), the output from the electromagnetic flowmeter should be 1 (v). When a given time t1 has elapsed, however, the output from the electromagnetic flowmeter may become 1.2 (v) in spite of the fact that the flow velocity of the fluid to be measured remains 1 (m/sec). A span shift can be considered as a cause of this output variation. A phenomenon called a span shift occurs when, for example, the value of an exciting current flowing in the exciting coil cannot be maintained constant due to a change of an ambient temperature of the electromagnetic flowmeter.